Xu, Hongkun A random fixed point theorem for multivalued nonexpansive operators in uniformly convex Banach spaces. (English) Zbl 0808.47044 Proc. Am. Math. Soc. 117, No. 4, 1089-1092 (1993). Summary: Let \((\Omega,\Sigma)\) be a measurable space with \(\Sigma\) a sigma-algebra of subsets of \(\Omega\), and let \(C\) be a nonempty, bounded, closed, convex, and separable subset of a uniformly convex Banach space \(X\). It is shown that every multivalued nonexpansive random operator \(T: \Omega\times C\to K(C)\) has a random fixed point, where \(K(C)\) is the family of all nonempty compact subsets of \(C\) endowed with the Hausdorff metric induced by the norm of \(X\). Cited in 10 Documents MSC: 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 60H25 Random operators and equations (aspects of stochastic analysis) Keywords:multivalued nonexpansive operator; measurable space; uniformly convex Banach space; random fixed point; nonempty compact subsets; Hausdorff metric × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Asuman G. Aksoy and Mohamed A. Khamsi, Nonstandard methods in fixed point theory, Universitext, Springer-Verlag, New York, 1990. With an introduction by W. A. Kirk. · Zbl 0713.47050 [2] A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), no. 5, 641 – 657. · Zbl 0339.60061 [3] A. T. Bharucha-Reid, Random integral equations, Academic Press, New York-London, 1972. Mathematics in Science and Engineering, Vol. 96. · Zbl 0327.60040 [4] Michael Edelstein, The construction of an asymptotic center with a fixed-point property, Bull. Amer. Math. Soc. 78 (1972), 206 – 208. · Zbl 0231.47029 [5] Kazimierz Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Mariae Curie-Skłodowska Sect. A 29 (1975), 69 – 72 (1977) (English, with Polish and Russian summaries). · Zbl 0365.47032 [6] Shigeru Itoh, A random fixed point theorem for a multivalued contraction mapping, Pacific J. Math. 68 (1977), no. 1, 85 – 90. · Zbl 0335.54036 [7] Shigeru Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261 – 273. · Zbl 0407.60069 · doi:10.1016/0022-247X(79)90023-4 [8] John L. Kelley, General topology, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. · Zbl 0066.16604 [9] W. A. Kirk, An iteration process for nonexpansive mappings with applications to fixed point theory in product spaces, Proc. Amer. Math. Soc. 107 (1989), no. 2, 411 – 415. · Zbl 0691.47043 [10] W. A. Kirk and Silvio Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990), no. 3, 357 – 364. · Zbl 0729.47053 [11] W. A. Kirk and Carlos Martínez-Yañez, Nonexpansive and locally nonexpansive mappings in product spaces, Nonlinear Anal. 12 (1988), no. 7, 719 – 725. · Zbl 0669.47032 · doi:10.1016/0362-546X(88)90024-7 [12] Teck Cheong Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123 – 1126. · Zbl 0297.47045 [13] Teck Cheong Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179 – 182 (1977). · Zbl 0346.47046 [14] Teck Cheong Lim, Characterizations of normal structure, Proc. Amer. Math. Soc. 43 (1974), 313 – 319. · Zbl 0284.47031 [15] Tzu-Chu Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1129 – 1135. · Zbl 0676.47041 [16] Zdzisław Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591 – 597. · Zbl 0179.19902 [17] Hong Kun Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), no. 2, 395 – 400. · Zbl 0716.47029 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.